Chemical Kinetics § Chemical kinetics deals with the study of elementary

reactions, determination of their rates, and development of reaction mechanisms. It is a specialized field of physical chemistry.

§ On the theoretical side, it involves fundamental analysis based on reaction rate theories and semi-empirical approaches, as well as more advanced theories based on quantum chemisty, for developing the appropriate reaction mechanisms. On the experimental side, it deals with the determination of reaction rate constants and the development of mechanisms using experimental data.

§ During the last four decades, there has been remarkable progress in this field, as reasonably reliable and detailed mechanisms have been developed for H2, CO, and several hydrocarbon and renewable fuels for a range of conditions in terms of pressure, equivalence ratio, etc.

§ It remains an active area of research, especially due to our interest in developing advanced predictive capabilities for reacting flows, and kinetic models for both conventional and renewable fuels.

Chemical Kinetics § Reacting flows are characterized by several time scales, for example,

reaction time, convection or flow time, diffusion time. § The ratio of transport time to reaction time is defined as the Damkohler

number (Da), one of the most important parameters in the study of combustion.

§ For Da << 1 (chemically frozen limit); the effect of kinetics may be neglected in some flow regime or domain.

§ Da >> 1 (equilibrium limit), the effect of finite rate chemistry may be neglected or circumvented in the numerical/ theoretical analysis. This leads to significant simplification in the analysis, for example, diffusion flames that are far from extinction.

§ However, the chemical time and transport time are often of the same order of magnitude, and the inclusion of finite rate chemistry is essential in the analysis of reacting flows.

Law of Mass Action Law of mass action provides a rate expression for any given reaction. Any elementary reaction can be written in a general form as:

For elementary reactions represented by Eq. (1) or (2), the mass conservation yields

€

υ i#

i=1

NS

∑ Mi = υ i##

i=1

NS

∑ Mi (1)

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H2 + 0.5O2 ⇒ H2O (3)

Here ν is the stoichiometry coefficient, and M a symbol for chemical species. The subscript i=1, 2,..Ns, with Ns being the total number of species involved in the reaction. It is important to distinguish between elementary and global reactions. Reactions 2a and 2b are elementary reactions, since they occur at the molecular level, while reaction (3) is a global reaction as it does not occur at the molecular level. Chemical kinetics deals with determining rates of both types of reactions.

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dN1

dt=dN2

dt.......=

dNi

dt= (υ i

## −υ i#) dξdt

(4)

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O2 + H⇒ OH +O (2a) O+H2 ⇒OH +H (2b)

Phenomenological Law of Mass Action Here Ni is the number of moles of species i, and ξ is the reaction progress variable indicating the degree of advancement of reaction. Dividing by the system volume, we can write the above equation in terms of the rate of change of concentration of each species as

k is defined as the reaction rate constant, and is generally expressed as

Here ω is the reaction rate in moles/volume/time. Based on experimental and theoretical studies, the phenomenological law of mass rate expression has been developed, which states that the reaction rate (ω) is proportional to the product of the concentrations of reactants.

(6)

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ω = k.Π iNs ci

$ υ i = k.c1$ υ 1c2

$ υ 2 ...

(5)

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dc1

dt=dc2

dt.......=

dcidt

= ˙ ω i = (υ i$$ −υ i

$) dξdt

1V

= (υ i$$ −υ i

$)ω

Here A is the pre-exponential factor, E the activation energy, Ru the universal gas constant, and T the temperature (K). Equation 7 is known as the Arrhenius equation.

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k = ATαe−(E /RuT ) (7)

Multiple Reactions Fuel oxidation chemistry generally involves a number of elementary reactions, which can be expressed by rewriting the equation (1) for the jth reaction as

Here j=1,2,..NR, with NR representing the total number of reactions. Then the rate of consumption (or production) of species i (moles/vol/time) is

(8)

€

υ i, j#

i=1

NS

∑ Mi = υ i, j##

i=1

NS

∑ Mi

(9)

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ω i =dcidt

= (υ i, j$$ −υ i, j

$)ω jj=1

NR

∑ with

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ω j = k j .Π iNS ci

$ υ i , j

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−d H[ ]dt

= −d O2[ ]dt

=d OH[ ]dt

=ω = k O2[ ] H[ ]

Thus for reaction (2a), one can write

§ Thus the rate of consumption (or production) of any species can be determined if the reaction mechanism and the reaction rate constant for each reaction in the mechanism are known.

§ These equations represent the source terms in the species and energy equations, which will be discussed later.

and the consumption or production rate of species (i) is

Then the net forward reaction rate for the jth reaction is

(11)

€

ω j = k j , f .Π iNS ci

$ υ i , j − k j ,b .Π iNS ci

$ $ υ i , j€

υ i, j#

i=1

NS

∑ Mi ⇔ υ i, j##

i=1

NS

∑ Miwhere j=1,2,..NR/2 (10)

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ω i = (υ i, j$$ −υ i, j

$)ω jj=1

NR / 2

∑ (12)

Forward and Backward Reactions

In a reacting system, reactions (for example, reaction (2)) can proceed in both forward and backward directions. Then equation (8) should be written as

     1.  h+o2=o+oh                                                                                    2.  o+h2=h+oh                                                                                    3.  oh+h2=h+h2o                                                                                4.  o+h2o=oh+oh        5.  h2+m=h+h+m        6.  o2+m=o+o+m        7.  oh+m=o+h+m                                                                                  8.  h2o+m=h+oh+m                                                                              9.  h+o2(+m)=ho2(+m)                                                                    10.  ho2+h=h2+o2                                                                              11.  ho2+h=oh+oh                                                                              12.  ho2+o=oh+o2                                                                              13.  ho2+oh=h2o+o2                                                                          14.  h2o2+o2=ho2+ho2                                                                      15.  h2o2+o2=ho2+ho2                                                                      16.  h2o2(+m)=oh+oh(+m)                                                                17.  h2o2+h=h2o+oh                                                                          18.  h2o2+h=h2+ho2                                                                          19.  h2o2+o=oh+ho2                                                                          20.  h2o2+oh=h2o+ho2                                                                      21.  h2o2+oh=h2o+ho2                                                                  

Reaction Mechanism for H2-O2

Reaction, NR=21 Species: Ns=7 H, O2, O, OH, H2, H2O, H2O2

€

υ i, j#

i=1

NS

∑ Mi = υ i, j##

i=1

NS

∑ Mi

j=1, 2, … NR

€

ω i =dcidt

= (υ i, j$$ −υ i, j

$)ω jj=1

NR

∑

€

ω j = k j , f .Π iNS ci

$ υ i , j − k j ,b .Π iNS ci

$ $ υ i , j

For practice, write equations for the rate of reaction # 3 and 5, and for the net rate of production of H2O

Collision Theory of Reaction Rates

Since reactions at the molecular level involve collisions between molecules, the reaction rate for a bimolecular reaction should be proportional to the collision frequency multiplied by the probability that a collision leads to reaction. The probability implies that only those molecules that possess energy greater than a certain threshold value will react, and is proportional to exp(-E/ RuT).

The collision frequency is determined using a simplified model based on ideal gas assumption. As depicted in the figure, the model considers two reactant species (say) A and B. It assumes that colliding molecules only possess translational energy. Let us consider the motion of one molecule (species A). It will collide with all B molecules that are contained in the cylindrical volume (V) swept by this molecule. Then the number of collisions encountered by this molecule is

€

ZA =VnB = πσ AB2 UABnB

Here nA is the number density of A molecules, σAB (cylinder radius) =0.5.(σA+σB), and UAB is the average velocity of A molecules with respect to B molecules.

σA σB σA + σB

UAB

Collision Theory of Reaction Rates

Considering nA as the number density of A molecules, the collision frequency can be written as

€

Z = πσAB2 UABnBnA = πσ AB

2 UABCBCAAv2

(1)

Collision Theory of Reaction Rates

Then the reaction rate can be written as

€

˙ ω = (Z /Av )e−(E /RuT ) = πσAB

2 UABCBCAAve−(E /RuT )P

Here exponent factor represents the probability of collision having sufficient energy (or proportion of collisions with energy greater that E) for reaction, and P is defined as the steric factor that accounts for the geometry of collisions, i.e., colliding molecules require certain orientation for reaction. UAB is given by (assuming Maxwell velocity distribution)

€

UAB2 =

8kbTπµAB

with µAB =mAmB

mA + mB

€

˙ ω =σAB2 (π8kbT /µAB )1/ 2CBCAAve

−(E /RuT )P

Then the reaction rate can be written as

Comparing this expression for the reaction rate with that given by the law of mass action yields an expression for the reaction rate constant as

(2)

(3)

(4)

(moles/m3/s)

Collision Theory of Reaction Rates

P and E (activation energy) are determined from more advanced theories. Note as discussed earlier, based on the law of mass action, the reaction rate is also be expressed as

€

k(T) =σAB2 (π8kbT /µAB )

1/ 2Ave−(E /RuT )P (5)

€

˙ ω = ATαCBCAe−(E /RuT )

Then the reaction rate constant can be written as

(6)

€

k(T) = ATαe−(E /RuT ) =σAB2 (π8kbT /µAB )

1/ 2Ave−(E /RuT )P (7)

This yields an expression for the preexponnetial factor (A) in terms of the molecular properties. Also it gives α=0.5, which may not be correct for many reactions.

Equilibrium Constants Since any reaction can proceed in both forward and backward directions, reaction (2) can be rewritten as

(13) Then the net forward rate for reaction (13) is

€

O2 + H⇔OH +O

If the reaction is in equilibrium, then the equilibrium constant for this reaction can be defined in terms of concentrations as

€

˙ ω = k fΠ iNS ci

$ υ i − kbΠ iNS ci

$ $ υ i (14)

(15)

€

Kc =k fkb

=Π iNS ci,e

( # # υ i − # υ i )

Since equilibrium constant can be determined experimentally and with better accuracy, it is commonly used to calculate the backward reaction rate constant. Thus

€

˙ ω = k f Π iNS ci

$ υ i −Kc−1Π i

NS ci$ $ υ i( ) (14a)

In some cases, the backward reaction rate is much slower than the forward rate, especially when the product species are radical species (see Eq. 13), whose concentrations are usually small.

Various Equilibrium Constants Kc can be related to the other two equilibrium constants, defined in terms of mole fractions and partial pressures, as

(18)

In writing the above equations, we have used pi=ciRuT and xi = pi/p. Further Kx and Kp are related through

(17)

€

Kp = Kx (p / po) " " υ i − " υ i∑€

Kx =Π iNS xi,e

( # # υ i − # υ i )

€

Kp =Π iNS (pi / p

o)( # # υ i − # υ i )

Use above relations for reaction (13) and for

€

O2 ⇔ 2O

Here Kx and Kp (recall the discussion of equilibrium constant based on species partial pressures) are defined as

€

Kc = Kp (po /RuT)

" " υ i − " υ i∑

€

Kc = Kx (p /RuT)" " υ i − " υ i∑

(19)

(16)

€

CO+ H2O⇔CO2 + H2

§ In a reacting flows, equations (9) provide the source (sink) terms in the conservation equations for species and energy. Thus if the reaction mechanism and reaction rate constants are known, it provides a closed system for direct numerical simulations (DNS) of chemically reacting flows.

§ However, development of a comprehensively validated mechanism along with the reaction rate constants has been one of the major challenges in simulating reacting flows. Even if a reliable mechanism is available, DNS of reacting flows has been an extremely challenging problem due to (i) the size of the mechanism, and (ii) a wide range of relevant length and time scales that are needed to be resolved.

§ Consequently, various approximations or assumptions are invoked to obtain an approximate but realistic solution. Using these approximations, such as steady state and partial equilibrium assumptions, a detailed mechanism can be simplified to obtain a reduced mechanism.

§ Depending upon the system complexity, the fuel oxidation chemistry may be represented by a relatively small number of reactions, i.e., a skeletal or semi-global mechanism. In the extreme case, the chemistry may be represented by a single reaction.

Steady State Approximation and Partial Equilibrium

Fuel oxidation chemistry generally involves number of reactions and species. Many of these species are radical species which play the role of intermediates or chain carriers. While some of these radical species are continuously being produced and consumed at rapid rates, their concentrations as well as their net rate of change of concentration is small. For example,

Steady State Approximation

€

H2 +OH⇔H2O+ H

The net rate of change of H concentration is small compared to its rate of production and consumption

€

dcidt

= ˙ ω i << k fΠ iNS ci

$ υ i or kbΠ iNS ci

$ $ υ i (20)

€

k fΠ iNS ci

# υ i = kbΠ iNS ci

# # υ i (21) Then

This reduces the differential equation to an algebraic equation for the solution of ci and thus reducing the overall computational effort.

Note: Above approximation does not mean that the concentration of ci is not changing, rather its rate of change is small relative to its production and consumption rates.

The assumption implies that in a multi-reaction scheme, the net reaction rate of (say) reaction j is small compared to its forward and backward reaction rates. Thus

Partial Equilibrium Assumption

§ Note the difference between this and the the steady state approximation, as the latter assumes that the rate of change of a particular species concentration is small.

§ It reduces the number of reactions in the mechanism, and provides an equation for the concentration of a particular species to be solved in terms of other species concentrations.

(22a)

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ω j << k j, f .Π iNS ci

$ υ i , j or k j,b .Π iNS ci

$ $ υ i , j

(22b)

€

k j , f .Π iNS ci

# υ i , j ≈ k j ,b .Π iNS ci

# # υ i , j

Chemical Kinetics

Elementary Reactions versus Global Reactions

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O2 + H⇔OH +O

Types of Reactions §  Unimolecular §  Bimolecular §  Termolecular

€

2H2 +O2 ⇒ 2H2O

Elementary Reaction Global Reaction

Global Reaction Rate and Reaction Order

€

˙ ω = Ae−(E /RuT ) H2[ ]a O2[ ]b A, E, a and b are determined through curve fitting with experimental data

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H2 ⇔H + H

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H +O2 ⇔OH +O

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H +O2 + M⇔HO2 + M